Calculator Input
Enter derivative values at a chosen parameter point. Third derivative fields are optional. Fill them to compute torsion too.
Example Data Table
| Curve | t | r′(t) | r″(t) | r‴(t) | Unit Binormal | Curvature | Torsion |
|---|---|---|---|---|---|---|---|
| r1(t) | 1 | (1, 2, 1) | (0, 1, 2) | (1, 0, 1) | (0.801784, -0.534522, 0.267261) | 0.254588 | 0.285714 |
| r2(t) | 0 | (2, 0, 0) | (0, 1, 0) | (0, 0, 1) | (0.000000, 0.000000, 1.000000) | 0.250000 | 0.500000 |
Formula Used
Let v = r′(t) and a = r″(t).
Cross product: v × a = (vyaz − vzay, vzax − vxaz, vxay − vyax)
Unit binormal: B = (v × a) / |v × a|
Unit tangent: T = v / |v|
Principal normal: N = B × T
Curvature: κ = |v × a| / |v|3
Torsion: τ = ((v × a) · r‴(t)) / |v × a|2
If |v × a| = 0, the unit binormal is undefined at that point.
How to Use This Calculator
- Enter a curve label for reference.
- Enter the parameter value if you want it shown in the result.
- Fill the first derivative components x′(t), y′(t), and z′(t).
- Fill the second derivative components x″(t), y″(t), and z″(t).
- Optionally add the third derivative components to compute torsion.
- Press the calculate button.
- Read the result section above the form.
- Use the CSV or PDF buttons to export your summary.
Unit Binormal Guide
What the unit binormal shows
The unit binormal vector describes the local orientation of a space curve. It is part of the Frenet frame. The frame also includes the unit tangent and the principal normal. Together they explain how a curve moves and twists in three dimensions. This is useful in calculus, geometry, CAD work, robotics, animation, and trajectory analysis. A correct unit binormal helps you detect turning direction and spatial rotation. It also confirms whether your derivative data produces a stable geometric frame.
Why derivative inputs matter
This calculator uses derivative values at a selected parameter point. The first derivative gives the local direction of motion. The second derivative helps measure bending. Their cross product points in the binormal direction. When that cross product has zero magnitude, the curve is locally straight or degenerate. In that case, the unit binormal is not defined. The calculator checks this automatically. It also reports speed, curvature, and orthogonality so you can verify your numbers before using them in a proof or model.
How the extra outputs help
The result is more than one vector. You also get the cross product, its magnitude, the unit tangent, the principal normal, and curvature. If you enter the third derivative, the calculator estimates torsion too. Torsion tells you how strongly the curve leaves its osculating plane. These extra outputs turn the page into a compact Frenet frame tool. They reduce manual work and help students compare related quantities without repeating the same derivative substitutions several times.
When to use this page
Use this page when you already know derivative values from a parametric curve. It works well for homework checks, lecture preparation, worked examples, and engineering math notes. The example table shows sample inputs and outputs. The export buttons make it easy to save a result summary for class records or printed review sheets. Because the layout is simple and direct, you can focus on the geometry, the formulas, and the meaning of each number.
FAQs
1. What is a unit binormal vector?
It is a unit vector perpendicular to both the tangent and principal normal. It completes the Frenet frame for a smooth space curve.
2. What inputs are required?
You need the three first derivative components and the three second derivative components at one parameter value. Third derivative inputs are optional.
3. Why can the unit binormal be undefined?
It becomes undefined when r′(t) × r″(t) equals zero. That means the local turning information is missing or the curve is degenerate at that point.
4. Do I need the third derivative?
No. The third derivative is only needed if you also want torsion. The unit binormal itself only needs the first and second derivatives.
5. How is curvature related to the result?
Curvature measures how sharply the curve bends. It uses the same derivative data, so it is natural to show it beside the unit binormal.
6. Can I use values from any parametric curve?
Yes. As long as you evaluate the derivatives at one parameter point and enter the numeric components, the calculator can process them.
7. What do the orthogonality checks mean?
They test whether T, N, and B are perpendicular as expected. Values near zero confirm the frame is numerically consistent.
8. What does the determinant check show?
It checks whether the computed frame behaves like a right-handed orthonormal basis. A value near one is the expected result.